aa r X i v : . [ m a t h . AG ] O c t Relating two genus 0 problems of John Thompson
Michael D. Fried
Abstract.
Excluding a precise list of groups like alternating, symmetric,cyclic and dihedral, from 1st year algebra ( § g =0 . We call the exceptional groups . Example: Finitely many Chevalley groups are 0-sporadic. Aproven result: Among polynomial reduced families . Each (miraculously) defines onenatural genus 0 Q cover of the j -line. The latest Nielsen class techniques applyto these dessins d’enfant to see their subtle arithmetic and interesting cusps.John Thompson earlier considered another genus 0 problem: To find θ -functions uniformizing certain genus 0 (near) modular curves. We call thisProblem g =0 . We pose uniformization problems for j -line covers in two cases.First: From the three 0-sporadic examples of Problem g =0 . Second: From finitecollections of genus 0 curves with aspects of Problem g =0 . Contents
1. Genus 0 themes1.1. Production of significant genus 0 curves1.2. Detailed results2. Examples from Problem g =0 H DP7 , H DP13 , H DP15 n = 13 and 154. j -line covers for polynomial PGL ( Z /
2) monodromy4.1. Branch cycle presentation and definition field4.2. b-fine moduli property5. j -line covers for polynomial PGL ( Z /
3) monodromy5.1. Degree 13 Davenport branch cycles ( g , g , g , g ∞ ) Mathematics Subject Classification.
Primary 11F32, 11G18, 11R58, 14G32; Secondary20B05, 20C25, 20D25, 20E18, 20F34.Thanks to NSF Grant
Thompson
Semester at U. of Florida. N = { Ni( G, C ) in } G ∈ Q F (2) × s H N = { Ni( G, C ± ) in } G ∈ Q F (3) × s H g =0 and Problem g =0
1. Genus 0 themes
We denote projective 1-space P with a specific uniformizing variable z by P z . This decoration helps track distinct domain and range copies of P . We useclassical groups: D n (dihedral), A n (alternating) and S n (symmetric) groups ofdegree n ; PGL ( K ), M¨obius transformations over K ; and generalization of theseto PGL u +1 ( K ) acting on k -planes, 0 ≤ k ≤ u −
1, of P u ( K ) ( K points of projective u -space). § P z by U =(( P z ) \ ∆ ) /S . For K a field, G K is its absolute Galois group (we infrequentlyallude to this for some applications).A g ∈ S n has an index ind( g ) = n − u where u is the number of disjoint cyclesin g . Example: (1 2 3)(4 5 6 7) ∈ S has index 7 − ϕ : X → P z is adegree n cover (of compact Riemann surfaces). We assume the reader knows aboutthe genus g X of X given a branch cycle description ggg = ( g , . . . , g r ) for ϕ ( § n + g X −
1) = P ri =1 ind( g i ) ([ V¨o96 , § Fr05 , Chap. 4]).
Compact Riemann sur-faces arose to codify two variable algebraic relations. The moduli of covers is arefinement. For a given genus, this refinement has many subfamilies, with associ-ated discrete invariants. Typically, these invariants are some type of
Nielsen class ( § g moduli with cases where the cover moduli has dimen-sion 1. The gains come by detecting the moduli resemblances to, and differencesfrom, modular curves.We emphasize: Our technique produces a parameter for families of equationsfrom an essential defining property of the equations. We aim for a direct descriptionof that parameter using the defining property. These examples connect two themesuseful for intricate work on families of equations. We refer to these as two genus 0problems considered by John Thompson. Our first version is a naive form.(1.1a) Problem g =0 : If a moduli space of algebraic relations is a genus 0 curve,where can we find a uniformizer for it?(1.1b) Problem g =0 : Excluding symmetric, alternating, cyclic and dihedral groups,what others are monodromy groups for primitive genus 0 covers?Our later versions of each statement explicitly connect with well-known problems.[ Fr80 ], [
Fr99 ], [
GMS03 ] show examples benefiting from the monodromy method . ENUS 0 PROBLEMS 3
We use the latest Nielsen class techniques ( § shift-incidencematrix from [ BFr02 , §
9] and [
FrS04 ]) to understand these parameter spaces asnatural j -line covers. They are not modular curves, though emulating [ BFr02 ] weobserve modular curve-like properties.Applying the Riemann-Roch Theorem to Problem g =0 does not actually answerthe underlying question. Even when (say, from Riemann-Hurwitz) we find a curvehas genus 0 that doesn’t trivialize uniformizing its function field. Especially whenthe moduli space has genus 0: We seek a uniformizer defined by the moduli problem.That the j -line covers of our examples have genus 0 allows them to effectively pa-rametrize (over a known field) solutions to problems with a considerable literature.We justify that — albeit, briefly — to give weight to our choices.[ FaK01 ] uses k -division θ -nulls (from elliptic curves) to uniformize certain mod-ular curves. Those functions, however, have nothing to do with the moduli for ourexamples. Higher dimensional θ -nulls on the (1-dimensional) upper-half plane areakin to, but not the same as, what quadratic form people call θ -functions. Theformer do appear in our examples of Problem g =0 ( § θ -confusing point). We are new at Monstrous Moonshine, though the requiredexpertise documented by [ Ra00 ] shows we’re not alone. Who can predict fromwhere significant uniformizers will arise? If a θ -null intrinsically attaches to themoduli problem, we’ll use it. § g -sporadic). All modular curves appear as(reduced; see § Fr78 , § j -line, responding to (1.1a), yet they are not modular curves.1.2.1. The moduli of three 0-sporadic monodromy groups.
Three polynomial 0-sporadic groups stand out on M¨uller’s list ( § n = 7 ,
13 and15, with four branch points (up to reduced equivalence § one (for each n ∈ { , , } )genus 0 j -line cover ( ψ n : X n → P j ) over Q . We tell much about these spaces, their b-fine moduli properties and their cusps (Prop. 4.1 and Prop. 5.1).We stress the uniqueness of ψ n , and its Q structure. Reason: The moduliproblem defining it does not produce polynomials over Q . Let K be the uniquedegree 4 extension of Q in Q ( e πi/ ). For n = 13, a parameter uniformizing X asa Q space gives coordinates for the four (reduced) families of polynomials over K .These appear as solutions of Davenport’s problem ( § Fr73 ], [
Fe80 ], [
Fr80 ] and[
Fr99 ]) suggested that genus 0 covers have a limited set of monodromy groups(Problem g =0 ). [ Fr99 , §
5] and [
Fr04 , App. D] has more on the applications.1.2.2.
Modular curve-like genus 0 and 1 curves.
Our second example is closerto classical modular curve themes, wherein uniformizers of certain genus 0 curvesappear from θ -functions. § Modular Tower , defined by the group F × s Z /
3, with F a free group on two generators. We call this the n = 3 case. Foreach prime p = 3, and for each integer k ≥
0, there is a map ¯ ψ p,k : ¯ H rd p,k → P j with¯ H rd p,k a (reduced) moduli space. The gist of Prop. 6.5: Each such ¯ H rd p,k is nonempty M. FRIED (and ¯ ψ p,k is a natural j -line cover). For a given p the collection { ¯ H rd k,p } k ≥ forms aprojective system; we use this below.We contrast the n = 3 case with the case F × s Z /
2. This is the n = 2 case: − ∈ {± } = Z / F to their inverses. We do this to give the Modular Tower view of noncomplex multiplication in Serre’s
Open Image Theorem ([ Se68 , IV-20]). The gist of Prop. 6.3: Serre’s Theorem covers less territory thanmight be expected. [
Fr04 , § exceptional covers. This shows Davenport’s problem is not an isolated example.The (strong) Main Conjecture on Modular Towers [ FrS04 , § n = 3. Only finitely many ¯ H rd p,k s have a genus 0 or 1 (curve) component.These are moduli spaces, and rational points on such components interpret signifi-cantly for many problems. For n = 2 the corresponding spaces are modular curves,and they have but one component. Known values of ( p, k ) where ¯ H rd p,k has morethan one component include p = 2, with k = 0 and 1. § j -line covers coming from Nielsenclasses there is a map from elements in the Nielsen class to cusps. The most mod-ular curve-like property of these spaces is that they fall in sequences attached to aprime p . Then, especially significant are the g- p ′ cusps ( § p ′ cusps are those we call Harbater-Mumford . For ex-ample, in this language the width p (resp. 1) cusp on the modular curve X ( p ) ( p odd) is the Harbater-Mumford (resp. shift of a Harbater-Mumford) cusp (Ex. 6.6).Prob. 6.7 is a conjectural refinement of Prop. 6.3. This distinguishes those compo-nents containing H-M cusps among the collection of all components of ¯ H rd p,k s.If right, we can expect applications for those components that parallel [ Se68 ](for n = 2). We conclude with connections between Problem g =0 and Problem g =0 .This gives an historical context for using cusps of j -line covers from Nielsen classes.(1.2a) Comparison of our computations with computer construction of equa-tions for the Davenport pair families in [ CoCa99 ] ( § g =0 and Problem g =0 ( §
2. Examples from Problem g =0 We briefly state Davenport’s problem and review Nielsen classes. Then, weexplain Davenport’s problem’s special place among polynomial 0-sporadic groups.
The name
Davenport pair (nowcalled S(trong)DP) first referred to pairs ( f, g ) of polynomials, over a number field K (with ring of integers O K ) satisfying this.(2.1) Range equality: f ( O /ppp ) = g ( O /ppp ) for almost all prime ideals ppp of O K .Davenport asked this question just for polynomials over Q . We also assume thereshould be no linear change of variables (even over ¯ K ) equating the polynomials.This is an hypothesis that we intend from this point. There is a complete descriptionof the Davenport pairs where f is indecomposable ( § i(sovalent)DP s: Each value in the range of f or g is achieved with the same multiplicity by both polynomials. As in [ AFH03 ,7.30], this completely describes all such pairs even with a weaker hypothesis: (2.1)holds for just ∞ -ly many prime ideals of O K . A Nielsen class is a combinatorial invariantattached to a (ramified) cover ϕ : X → P z of compact Riemann surfaces. If ENUS 0 PROBLEMS 5 deg( ϕ ) = n , let G ϕ ≤ S n be the monodromy group of ϕ . The cover is primitive or indecomposable if the following equivalent properties hold.(2.2a) It has no decomposition X ϕ ′ −→ W ϕ ′′ −→ P z , with deg( ϕ ′ ) ≥
2, deg( ϕ ′′ ) ≥ G ϕ is a primitive subgroup of S n .Let zzz be the branch points of ϕ , U zzz = P z \ { zzz } and z ∈ U zzz . Continue pointsover z along paths based at z , having the following form: γ · δ i γ − , γ, δ on U zzz and δ i a small clockwise circle around z i . This attaches to ϕ a collection of conjugacyclasses C = (C , . . . , C r } , one for each z i ∈ zzz . The associated Nielsen class :Ni = Ni( G, C ) = { ggg = ( g , . . . , g r ) | g · · · g r = 1 , h ggg i = G and ggg ∈ C } . Product-one is the name for the condition g · · · g r = 1. From it come invariantsattached to spaces defined by Nielsen classes. Generation is the name of condition h ggg i = G . Writing ggg ∈ C means the g i s define conjugacy classes in G , possibly inanother order, the same (with multiplicity) as those in C . So, each cover ϕ : X → P z has a uniquely attached Nielsen class: ϕ is in the Nielsen class Ni( G, C ).2.2.1. Standard equivalences.
Suppose we have r (branch) points zzz , and a cor-responding choice ¯ ggg of classical generators for π ( U zzz , z ) [ BFr02 , § G, C ) lists all homomorphisms from π ( U zzz , z ) to G . These give a cover withbranch points zzz associated to ( G, C ). Elements of Ni( G, C ) are branch cycle de-scriptions for these covers relative to ¯ ggg . Equivalence classes of covers with a fixedset of branch points zzz , correspond one-one to equivalence classes on Ni( G, C ). Wecaution: Attaching a Nielsen class representative to a cover requires picking onefrom many possible r -tuples ¯ ggg . So, it is not an algebraic process.[ BFr02 , § Q ′′ below. Let N S n ( G, C ) be those g ∈ S n normalizing G and permuting the collection of conjugacy classes in C . Absolute (resp. inner)equivalence classes of covers (with branch points at zzz ) correspond to the elementsof Ni( G, C ) /N S n ( G, C ) = Ni( G, C ) abs (resp. Ni( G, C ) /G = Ni( G, C ) in ). Especiallyin § absolute , inner and for each of these reduced equivalence. These showhow to compute specific properties of H ( G, C ) abs , H ( G, C ) in and their reducedversions, parametrizing the equivalences classes of covers as zzz varies.2.2.2. Reduced Nielsen classes.
Reduced equivalence corresponds each cover ϕ : X → P z to α ◦ ϕ : X → P z , running over α ∈ PGL ( C ). If r = 4, a nontrivialequivalence arises because for any zzz there is a Klein 4-group in PGL ( C ) mapping zzz into itself. (An even larger group leaves special, elliptic , zzz fixed.) This interpretsas an equivalence from a Klein 4-group Q ′′ acting on Nielsen classes ( § G, C ) / h N S n ( G, C ) , Q ′′ i = Ni( G, C ) abs , rd (resp.Ni( G, C ) / h G, Q ′′ i = Ni( G, C ) in , rd ) . These give formulas for branch cycles presenting H ( G, C ) abs , rd and H ( G, C ) in , rd as upper half plane quotients by a finite index subgroup of PSL ( Z ). This is a ram-ified cover of the classical j -line branching over the traditional places (normalizedin [ BFr02 , Prop. 4.4] to j = 0 , , ∞ ). Points over ∞ are meaningfully called cusps.Here is an example of how we will use these. § j -line covers (dessins d’enfant) conjugate over Q ( √−
7) parametrizing reduced classes of degree 7 Davenport polynomial pairs. Infact, the (by hand) Nielsen class computations show the covers are equivalent over Q . This same phenomenon happens for all pertinent degrees n = 7 , ,
15, though
M. FRIED the field Q ( √−
7) changes and corresponding Nielsen classes have subtle differences.You can see these by comparing n = 7 with n = 13 ( § Let u ≥
2. A
Singer cycle is agenerator α of F ∗ q u +1 , acting by multiplication as a matrix through identifying F u +1 q and F q u +1 . Its image in PGL u +1 acts on points and hyperplanes of P u ( F q ).Let G be a group with two doubly transitive representations T and T , equiv-alent as group representations, yet not permutation equivalent, and with g ∞ ∈ G an n -cycle in T i , i = 1 ,
2. Excluding the well-documented degree 11 case, G hasthese properties ([ Fe80 ], [
Fr80 ], [
Fr99 , § α the conjugacy class of g ∞ .(2.3a) G ≥ PSL u +1 ( F q ); T and T act on points and hyperplanes of P u .(2.3b) n = ( q u +1 − q −
1) and g ∞ is a Singer n -cycle.2.3.1. Difference sets.
Here is how difference sets ( § Definition . Call
D ≤ Z /n a difference set if nonzero differences from D distribute evenly over Z \ { } . The multiplicity v of the appearance of each elementis the multiplicity of D . Regard a difference set and any translate of it as equivalent.Given the linear representation from T on x , . . . , x n , the representation T ison { P i ∈D + j x i } nj =1 with D a difference set. The multiplier group M n of D is { m ∈ ( Z /n ) ∗ | m · D = D + j m , with j m ∈ Z /n } . We say m ·D is equivalent to D if m ∈ M n . In PGL u +1 ( F q ), α m is conjugate to α ex-actly when m ∈ M n . The difference set −D corresponds to an interchange betweenthe representations on points and hyperplanes. Conjugacy classes in PGL u +1 ( F q )of powers of α correspond one-one to difference sets equivalence classes mod n .2.3.2. Davenport pair Nielsen classes.
We label our families of polynomials byan m ∈ ( Z /n ) ∗ \ M n that multiplies the difference set to an inequivalent differenceset. Our families are of absolute reduced classes of covers in a Nielsen class. Conju-gacy classes have the form (C , . . . , C r − , C α ), the groups satisfy G ≥ PSL u +1 ( F q ),and covers in the class have genus 0. Two results of Feit show r − ≤ ≥ n/ u +1 ( F q ) [ Fe73 ].(2.4b) ind(C) ≥ n ( q − /q if C is a conjugacy class in PΓL u +1 ( F q ) [ Fe92 ].. Conclude: If n is odd, then r − n = 7: C i s are in the conjugacy class of transvections (fixing a hyper-plane), with index 2. So, they are all conjugate.(2.5b) n = 13: C i s are in the conjugacy class of elements fixing a a hyperplane(determinant -1), so they generate PGL ( Z /
3) and all are conjugate.(2.5c) n = 15: Two of the C i s are in the conjugacy class of transvections, andone fixes just a line.(2.5d) n = 7 , , r = 3.Transvections in GL u +1 have the form vvv vvv + µ H ( vvv ) vvv with µ H a linear functionalwith kernel a hyperplane H , and vvv ∈ H \ { } [ A57 , p. 160]. For q a power oftwo, these are involutions: exactly those fixing points of a hyperplane. For q odd,involutions fixing the points of a hyperplane (example, induced by a reflection inGL n +1 in the hyperplane) have the maximal number of fixed points. When q isa power of 2, there are involutions fixing precisely one line. Jordan normal form ENUS 0 PROBLEMS 7 shows these are conjugate to a b c . This is an involution if and only if ab = bc = 0. So, either b = 0 or a = c = 0. Onlyin the latter case is the fixed space a line. So, given the conjugacy class of g ∞ , onlyone possible Nielsen class defines polynomial Davenport pairs when n = 15. Wereprise M¨uller’s list of the polynomial 0-sporadic groups ([
Mu95 ]). Since sucha group comes from a primitive cover, it goes with a primitive permutation rep-resentation. As in § ( F q ) (for very small q ) and the Matthieu groups of degree 11 and 23. Then, all remaining groups fromhis list are from [ Fr73 ] and have properties (2.3). [
Fr99 , §
9] reviews and completesthis. These six polynomial 0-sporadic groups (with corresponding Nielsen classes)all give Davenport pairs. We concentrate on those three having one extra property:(2.6) Modulo PGL ( C ) (reduced equivalence as in § • They have degrees from { , , } and r = 4. • All r ≥ §
2) are in one of the respectively, 2, 4 or 2 Nielsen classes correspondingto the respective degrees 7, 13, 15.[
Fr73 ] outlines this. [
Fr80 , § H DP7 , H DP13 and H DP15 denote the spaces of polynomial covers that are onefrom a Davenport pair having four branch points (counting ∞ ). The subscriptdecoration corresponds to the respective degrees. We assume absolute, reducedequivalence (as in §
3. Explanation of the components for H DP , H DP , H DP The analytic families of respective degree n polynomials fall into several com-ponents. Each component, however, corresponds to a different Nielsen class. Forexample, H DP7 , the space of degree 7 Davenport polynomials has two components:with a polynomial associated to a polynomial in the other as a Davenport Pair.
Often we apply Nielsen classes to problemsabout the realization of covers over Q . Then, one must assume C is rational.[ Fr73 , Thm. 2] proved (free of the finite simple group classification) that no inde-composable polynomial DPs could occur over Q .There are polynomial covers in our Nielsen classes. So, ggg ∈ Ni( G, C ) has an n -cycle entry, g ∞ . These conjugacy classes for all n are similar: C n,u ; k ,k ,k = (C k , C k , C k , C ( α ) u ) , where C k denotes a (nontrivial) conjugacy class of involutions of index k and( u, n ) = 1. We explain the case n = 7 in the following rubric. M. FRIED (3.1a) Why C ki = C , j = 1 , ,
3, is the conjugacy class of a transvection(denote the resulting conjugacy classes by C ,u ;3 · ).(3.1b) Why the two components of H DP7 are H + = H (PSL ( Z / , C , · ) abs , rd and H − = H (PSL ( Z / , C , − · ) abs , rd . (3.1c) Why the closures of H ± over P j (as natural j -line covers) are equivalentgenus 0, degree 7 covers over Q .(3.1d) Why H ± , as degree 7 Davenport moduli, have definition field Q ( √− (3.1a) and (3.1b) . [ Fr73 , Lem. 4]normalizes Nielsen class representatives ( g , g , g , g ∞ ) for DP covers so that in bothrepresentations T j,n , j = 1 , g ∞ = (1 2 . . . n ) − identifies with some allowable α n .Regard ( g ∞ ) T ,n ( g ∞ in the representation T ,n ) as translation by -1 on Z /n .Then, ( g ∞ ) T ,n is translation by -1 on the collection of sets {D + c } c ∈ Z /n . Take v to be the multiplicity of D . Then, v ( n −
1) = k ( k −
1) with 1 < k = |D| < n −
1. InPGL u +1 ( F q ), α un is conjugate to α n if and only if u · D is a translation of D . Thatis, u is a multiplier of the design. Also, -1 is always a nonmultiplier [ Fr73 , Lem. 5].Here n = 7, so v ( n −
1) = k ( k −
1) implies k = 3 and v = 1. You find mod 7: D = { , , } and −D are the only difference sets mod translation. The multiplierof D is M = h i ≤ ( Z / ∗ : 2 · D = D + c ( c = 0 here).Covers in this Nielsen class have genus 0. Now use that in PSL ( Z / . So, 2 is the index of entries of ggg for all finite branchpoints. We have shown (3.1b) has the only two possible Nielsen classes. (3.1b) . We now show the two spaces H ± areirreducible, completing property (3.1b).Computations in [ Fr95a , p. 349] list absolute Nielsen class representatives with g ∞ the 4th entry. Label finite branch cycles ( g , g , g ) (corresponding to a poly-nomial cover, having g ∞ in fourth position) as Y , . . . , Y . There are 7 up to con-jugation by h g ∞ i , the only allowance left for absolute equivalence.(3.2) Y :((3 5)(6 7) , ((4 5)(6 2) , (3 6)(1 2)); Y :((3 5)(6 7) , (3 6)(1 2) , (3 1)(4 5)); Y : ((3 5)(6 7) , (1 6)(2 3) , (4 5)(6 2)); Y : ((3 5)(6 7) , (1 3)(4 5) , (2 3)(1 6)); Y :((3 7)(5 6) , (1 3)(4 5) , (2 3)(4 7)); Y :((3 7)(5 6) , (2 3)(4 7) , (1 2)(7 5)); Y :((3 7)(5 6) , ((1 2)(7 5) , (1 3)(4 5)) . The element (3 5)(6 7) represents a transvection fixing points of a line ⇐⇒ el-ements of D . Note: All entries in Y , . . . , Y of Table (3.2) correspond to transvec-tions. So these are conjugate to (3 5)(6 7). From this point everything reverts toHurwitz monodromy calculation with The elements q i , i = 1 , ,
3. Each acts by atwisting action on any 4-tuple representing a Nielsen class element. For example,(3.3) q : ggg ( ggg ) q = ( g , g g g − , g , g ) . n = 13 and 15. Up to translation there are 4 differencesets modulo 13. All cases are similar. So we choose D = { , , , } to bespecific. Others come from multiplications by elements of ( Z / ∗ . Multiplying by h i = M ( § g , g , g fixes all points of some line, and one extra point, a total offive points. Any column matrix A = ( vvv | eee | eee ) with vvv anything, and { eee i } i =1 the ENUS 0 PROBLEMS 9 standard basis of F , fixes all points of the plane P of vectors with 0 in the 1stposition. Stipulate one other fixed point in P ( F ) to determine A in PGL ( F ).Let ζ = e πi/ . Identify G ( Q ( ζ ) / Q ) with ( Z / (13)) ∗ . Let K be the fixedfield of M in Q ( ζ ). Therefore K is Q ( ζ + ζ + ζ ), a degree 4 extension of Q . Akin to when n = 7 take g ∞ = (1 2 . . .
12 13) − . The distinct difference sets(inequivalent under translation) appear as 6 j · D , j = 0 , , , Z / ∗ ).For future reference, though we don’t do the case n = 15 completely here, D = { , , , , , , } is a difference set mod 15. Its multiplicity is v in v (15 −
1) = k ( k −
1) forcing k = 7 and v = 3. The multiplier group is M = h i , sothe minimal field of definition of polynomials in the corresponding Davenport pairsis Q ( P j =0 ζ j ), the degree 2 extension of Q that √− generates. So, this case,like n = 7, has two families of polynomials appear as associated Davenport pairs.As with n = 7, use the notation C ( α ) u , u ∈ ( Z / ∗ for the conjugacy classesof powers of 13-cycles. Prop. 5.1 shows the following.(3.4a) Why C ki = C , i = 1 , ,
3, is the conjugacy class fixing all points of aplane (denote the resulting conjugacy classes by C ,u ;3 · ).(3.4b) Why the four components of H DP13 are H i = H (PGL ( Z / , C , i ;3 · ) abs , rd , i = 0 , , , . (3.4c) Why closure of all H i s over P j are equivalent genus 0, degree 13 coversover Q . Yet, H i , as degree 13 Davenport moduli, has definition field K . j -line covers for polynomial PGL ( Z / monodromy We produce branch cycles for the two j -line covers ¯ ψ ± : ¯ H ± → P j , i = 1 , j -line covers, though distinct as families of degree 7 covers. Our original nota-tion, H ± is for points of ¯ H ± not lying over j = ∞ . Each ppp + ∈ H + ( ¯ Q ) has acorresponding point ppp − ∈ H − ( ¯ Q ) denoting a collection of polynomial pairs { ( β ◦ f ppp + , β ◦ f ppp − ) } β ∈ PGL (¯ Q ) . The absolute Galois group of Q ( ppp + ) = Q ( ppp − ) maps this set into itself. Represen-tatives for absolute Nielsen class elements in Table (3.2) suffice for our calculation.This is because reduced equivalence adds the action of Q ′′ = h ( q q q ) , q q − i .This has the following effect.(4.1) Each ggg ∈ Ni + is reduced equivalent to a unique absolute Nielsen classrepresentative with g ∞ in the 4th position.Example: If ggg has g ∞ in the 3rd position, apply q − q to put it in the 4th position.[ BFr02 , Prop. 4.4] produces branch cycles for ¯ ψ ± . Reminder: The imagesof γ = q q and γ = q q q in h q , q , q i / Q ′′ = PSL ( Z ) identify with canonicalgenerators of respective orders 3 and 2. The product-one condition γ γ γ ∞ = 1 with γ ∞ = q holds mod Q ′′ . Compute that q with (3.3) action is q ∗ = (3 5 1)(4 7 6 2); q acts as q ∗ = (1 3 4 2)(5 7 6). So, γ acts as γ ∗ = (3 7 5)(1 4 6). From product-one, γ acts as γ ∗ = (3 6)(7 1)(4 2).Denote the respective conjugacy classes of ( γ ∗ , γ ∗ , γ ∗∞ ) = γγγ ∗ in the group theygenerate by C ∗ = (C , C , C ∞ ). Denote P j \ {∞} by U ∞ . Proposition . The group G = h γ ∗ , γ ∗ , γ ∗∞ i is S . Then, γγγ ∗ representsthe only element in Ni( S , C ∗ ) ′ : absolute equivalence classes with entries, in order,in the conjugacy class C ∗ . So, there is a unique cover ¯ ψ : X = X → P j in Ni( S , C ∗ ) ′ (ramified over { , , ∞} ). Restricting over U ∞ gives ψ : X ∞ → U ∞ equivalent to H ± → U ∞ of (3.1b) : It parametrizes each of the two absolute reducedfamilies of P z covers representing degree 7 polynomials that appear in a DP.The projective curve X has genus 0. It is not a modular curve. The spaces H ± are b-fine (but not fine; § Q ( √− . Their correspondingHurwitz spaces are fine moduli spaces with a dense set of Q ( √− points. As acover, however, ¯ ψ has definition field Q , and X has a dense set of Q points. Proof.
Since the group G is transitive of degree 7, it is automatically primi-tive. Further, ( γ ∗∞ ) is a 3-cycle. It is well-known that a primitive subgroup of S n containing a 3-cycle is either A n or S n . In, however, our case γ ∗∞ A , so G = S .We outline why Ni( S , C ∗ ) ′ has but one element. The centralizer of γ ∗∞ is U = h (1 3 4 2) , (5 7 6) i . Modulo absolute equivalence, any ( g , g , g ∞ ) ∈ Ni( S , C ∗ ) ′ has γ ∗∞ in the 3rd position. Let F = { , } and let x i be the fixed element of g i , i = 0 ,
1. Elements of F represent the two orbits O , O of γ ∗∞ (2 ∈ O and 5 ∈ O ).Conjugating by elements of U gives four possibilities:(4.2a) (*) x = 2 and x ∈ O \ { } ; or (**) x = 5 and x ∈ O \ { } ; or(4.2b) (*) x = 5 and x = 2; or (**) x = 2 and x = 5.We show the only possibility is (4.2b) (**), and for that, there is but oneelement. First we eliminate (4.2a) (*) and (**). For (4.2a) (**), then x = 5 and x = 7 or 6. The former forces (up to conjugation by U ) g = (7 6 1) · · · , g = (5 6) · · · . Then, g γ ∗∞ fixes 6, contradicting (6) g = 1. Also, x = 6 fails. Consider (4.2a) (*),so x = 1 or 3 ((2 4) appears in g automatically). Symmetry between the casesallows showing only x = 1. This forces (1 4 3) in g and h γ ∗∞ , g i is not transitive.Now we eliminate (4.2b) (*). Suppose x = 5, x = 2. This forces either g = (6 1 2) · · · , g = (5 6)(1 7) · · · or g = (6 x ?)(1 2 y ) , g = (5 6)( x y · · · . In the first, 4 → y in the latter.Conclude x = 2, x = 5. Previous analysis now produces but one possibleelement in Ni( S , C ∗ ) ′ . Applying Riemann-Hurwitz, using the index contributionsof g ∗ , g ∗ , g ∗∞ in order, the genus of the cover as g satisfies 2(7 + g −
1) = 3 + 4 + 5.So, g = 0.That X is not a modular curve follows from WohlFahrt’s’s Theorem [ Wo64 ].If it were, then its geometric monodromy group would be a quotient of PSL ( Z /N ): N is the least common multiple of the cusp widths. So, ord( γ ∗∞ ) = N = 12. Since,however, PSL ( Z /
12) = PSL (3) × PSL (4), it does not have S as a quotient.We account for the b-fine moduli property. This is equivalent to Q ′′ acting faith-fully on absolute Nielsen classes [ BFr02 , Prop. 4.7]. This is so from (4.1), nontrivialelements of Q ′′ move the conjugacy class of the 7-cycle. [ BFr02 , Prop. 4.7] alsoshows it is a fine reduced moduli space if and only if γ ∗ and γ ∗ have no fixed points.Here both have fixed points. So, the spaces H ± are not fine moduli spaces. (cid:3) ENUS 0 PROBLEMS 11
We explain [
BFr02 , § P having fine moduli over a field K has this effect. For ppp ∈ P corresponding to a specific algebraic object up to iso-morphism, you have a representing object with equations over K ( ppp ) with K . (Wetacitly assume P is quasiprojective — our H and H rd spaces are actually affine— to give meaning to the field generated by the coordinates of a point.)4.2.1. The meaning of b-fine.
The b in b-fine stands for birational . It meansthat if ppp ∈ H rd is not over j = 0 or 1, the interpretation above for ppp as a fine modulipoint applies. It may apply if j = 0 or 1, though we cannot guarantee it.If P is only a moduli space (not fine), then ppp ∈ P may have no representingobject over K ( ppp ) (and certainly can’t have one over a proper subfield of K ( ppp )).Still, G K ( ppp ) stabilizes the complete set of objects over K ( ppp ) representing ppp .For any Nielsen class of four branch point covers, suppose the absolute (notreduced) space H has fine moduli. The condition for that is no element of S n centralizes G . That holds automatically for any primitive non-cyclic group G ≤ S n (so in our cases; all references on rigidity in any form have this). For ppp , denote theset of points described as the image of ppp in U = (( P z ) \ ∆ ) /S by zzz ppp . Assume K isthe field of rationality of the conjugacy classes. Conclude: There is a representativecover ϕ ppp : X ppp → P z branched over zzz ppp and having definition field K ( ppp ).4.2.2. Producing covers from the b-fine moduli property.
Use the notation H abs for nonreduced space representing points of the Nielsen class. Then H ± identifieswith H abs ± / PGL ( C ). Consider our degree 7 Davenport pair problem and theirreduced spaces. The Nielsen classes are rational over K = Q ( √−
7) (Prop. 4.1).Any ppp rd ∈ H ± represents a cover ϕ ppp rd : X ppp rd → Y ppp rd with Y ppp rd a conic in P [ BFr02 ,Prop. 4.7] and ϕ ppp rd degree 7. Often, even for b-fine moduli and ppp rd not over 0 or 1,there may be no ppp ∈ H abs lying over ppp rd with K ( ppp ) = K ( ppp rd ). Such a ppp would give ψ ppp : X ppp → P z over K ( ppp rd ) representing ϕ ppp rd .Yet, in our special case, X ppp rd has a unique point totally ramified over Y ppp rd . Itsimage in Y ppp rd is a K ( ppp rd ) rational point. So Y ppp rd is isomorphic to P z over K ( ppp rd ).Up to an affine change of variable over K , there is a copy of P = X over K that parametrizes degree 7 Davenport pairs (over any nontrivial extension L/K ). j -line covers for polynomial PGL ( Z / monodromy We start by listing the 3-tuples ( g , g , g ) for X i in Table (5.1) that representan absolute Nielsen class representative by tacking g ∞ = (1 · · · − on the end[ Fr99 , § ( g , g , g , g ∞ ) . The followingelements are involutions fixing the hyperplane corresponding to the difference set.Each fixes one of the 9 points off the hyperplane. Compute directly possibilities for g , g , g since each is a conjugate from this list:(7 8)(5 11)(6 12)(9 13); (3 11)(7 13)(6 8)(9 12); (3 12)(5 8)(7 9)(11 13);(5 13)(6 9)(11 12)(3 8); (5 6)(3 7)(8 11)(12 13); (6 7)(8 11)(5 12)(3 13);(3 5)(7 12)(6 13)(8 9); (3 6)(5 9)(7 11)(8 13); (5 7)(6 11)(8 13)(3 9) . Here is the table for applying the action of q , q , q . (5.1) X :(6 7)(8 11)(5 12)(3 13) , (2 3)(13 4)(6 8)(9 10) , (1 2)(13 5)(6 12)(9 11) X :(6 7)(8 11)(5 12)(3 13) , (1 2)(13 5)(6 12)(9 11) , (1 3)(5 4)(12 8)(11 10) X :(3 5)(7 12)(6 13)(8 9) , (1 6)(2 3)(13 7)(12 10) , (8 10)(12 11)(6 2)(5 4) X :(3 5)(7 12)(6 13)(8 9) , (8 10)(12 11)(6 2)(5 4) , (1 2)(6 3)(13 7)(11 8) X :(5 6)(3 7)(9 11)(12 13) , (1 3)(4 5)(8 12)(11 10) , (2 3)(7 4)(1 8)(12 9) X :(5 6)(3 7)(9 11)(12 13) , (9 12)(2 3)(7 4)(1 8) , (8 2)(7 5)(1 9)(11 10) X :(5 6)(3 7)(9 11)(12 13) , (1 9)(2 8)(7 5)(10 11) , (4 5)(3 8)(9 2)(12 1) X :(5 6)(3 7)(9 11)(12 13) , (4 5)(3 8)(9 2)(12 1) , (12 2)(9 3)(7 4)(11 10) X :(8 9)(7 12)(13 6)(3 5) , (1 2)(6 3)(13 7)(11 8) , (1 3)(5 4)(12 8)(11 10) X :(6 7)(8 11)(5 12)(3 13) , (1 3)(4 5)(12 8)(11 10) , (6 8)(10 9)(4 13)(3 2) X :(7 3)(5 6)(12 13)(9 11) , (1 2)(7 5)(3 12)(8 9) , (1 3)(5 4)(12 8)(11 10) X :(5 6)(3 7)(9 11)(12 13) , (10 11)(2 12)(3 9)(7 4) , (1 2)(3 12)(7 5)(9 8) X :(8 9)(6 13)(7 12)(3 5) , (1 3)(5 4)(12 8)(11 10) , (2 3)(1 6)(7 13)(10 12) . Here are the j -line branch cycle descriptions. From action (3.3) on Table (5.1)(5.2) γ ∗ = q q = (1 5 3)(6 9 13)(2 8 11)(4 7 10) γ ∗ = q q q = (1 4)(2 5)(3 6)(7 9)(8 10)(11 12) ,γ ∗∞ = q = (1 10 2)(3 13 9 4)(5 11 12 8 7 6) . Again, you figure γ ∗ from the product one condition. In analogy to Prop. 4.1we prove properties (3.4) for degree 13 Davenport pairs. Denote h γ ∗ , γ ∗ , γ ∗∞ i by G . Proposition . Then, G = A . With C ∗ = (C , C , C ∞ ) the conjugacyclasses, respectively, of γγγ ∗ , Ni( A , C ∗ ) ′ (absolute equivalence classes with entriesin order in C ∗ ) has one element. So, there is a unique cover ¯ ψ : X = X → P j representing it (ramified over { , , ∞} ). Restrict over U ∞ for ψ : X ∞ → U ∞ equivalent to each H j → U ∞ , j = 0 , , , .The projective curve X has genus 0 and is not a modular curve. The spaces H j are b-fine (not fine) moduli spaces over K . Their corresponding Hurwitz spacesare fine moduli spaces with a dense set of K points. As a cover, however, ¯ ψ hasdefinition field Q , and X has a dense set of Q points. First compute the genus g of the curve in thecover presented by ¯ ψ to be 0, from2(13 + g −
1) = ind( γ ∗ ) + ind( γ ∗ ) + ind( γ ∗∞ ) = 4 · . Now we show why the geometric monodromy of the cover ¯ ψ is A . There arenine primitive groups of degree 13 [ Ca56 , p. 165]. Three affine groups Z /p × s U ;each U ≤ ( Z / ∗ , with U = { } , or having order 2 or order 3. Then, there are sixother groups S , A and PGL ( Z / , PSL ( Z /
3) with each of the last two actingon points and hyperplanes. The generators γ ∗ and γ ∗ are in A . It would be cuteif the monodromy group were PGL ( Z /
3) (the same as of the covers representedby its points). We see, however, ( γ ∗∞ ) fixes way more than half the integers in { , . . . , } . This is contrary to the properties of PSL ( Z / , PGL ( Z / A , C ∗ ) ′ . Like n = 7, assume triples of form ( g , g , γ ∗∞ ).To simplify, conjugate by an h ∈ S to change γ ∗∞ to (1 2 3 4)(5 6 7 8 9 10)(11 12 13)(keep its name the same). Its centralizer is U = h (1 2 3 4) , (5 6 7 8 9 10) , (11 12 13 14) i .Take F = { , , } as representatives, in order, of the three orbits O , O , O of γ ∗∞ on { , . . . , } . Suppose g i fixes x i ∈ { . . . , } , i = 0 ,
1. If x and x are in ENUS 0 PROBLEMS 13 different O k s, conjugate by U , and use transitivity of h g , g i , to assume x is oneelement of F and x another. We show the following hold.(5.3a) Neither g nor g fixes an element of O .(5.3b) The fixed point of g is in O or O , and of g in the other.(5.3c) With no loss we may assume either x = 5 and x = 1, or x = 1 and x = 5, and the latter is not possible.5.2.1. Proof of (5.3a) . Suppose x ∈ O ; with no loss conjugating by U takeit to be 11. Then, g = (12 13 y ) · · · and g = (11 13)( y · · · (using product-onein the form γ ∗∞ g g = 1). Then, however, g g fixes y : a contradiction, since γ ∗∞ fixes nothing. Now suppose x ∈ O ( x = 11). Then, g = (12 11 y ) · · · and g = ( y · · · . Transitivity of h γ ∗∞ , g i prevents y = 13. Conjugating by U allowstaking y = 1 or y = 5. If y = 1, then you find g = (12 11 1)(2 13 4) · · · and g = (1 13)(4 12)(2 3) · · · . Now, h γ ∗∞ , g i leaves O ∪ O stable, so is not transitive.5.2.2. Proof of (5.3b) , case x , x ∈ O . With no loss x = 1 and g = (1 4) · · · .This forces g = (2 4 z ) · · · , so under our hypothesis, (3) g = 3. This forces g =(1 4)(2 3) · · · and h γ ∗∞ , g i is stable on O .5.2.3. Proof of (5.3b) , case x , x ∈ O . With no loss x = 5 and g = (6 10 z ) · · · , g = (10 5)( z · · · . We separately show x ∈ { , , } are impossible, and by symmetry, x ∈ O is im-possible. If x = 6, then z = 7, and g = (6 10 7)(8 9 w ) · · · , g = (10 5)(7 9)( w · · · .The contradiction is that g g fixes w .If x = 7, then g = (6 10 z )(8 7 w ) · · · , g = (10 5)( z w · · · . So neither z nor w can be 8 or 9: With no loss w ∈ { , } . If w = 11, then with no loss z ∈ { , } . With z = 1, (4) g g = 11, a contradiction. If z = 12, g = (6 10 12)(8 7 11)(13 9 u ) · · · , g = (10 5)(12 9)(11 6)(8 13) · · · . Check: u must be 13, a contradiction.Finally, w = 1 forces g = (6 10 2)(8 7 1)(3 9 ?) · · · , g = (10 5)(2 9)(1 6)(8 4) · · · .So, (9) g g = 8 forces ? = 4, and that forces g to fix 3, a contradiction to x = 7.That leaves x = 8 and g = (6 10 z )(9 8 w ) · · · , g = (10 5)( z w · · · . Con-jugate by U to assume w ∈ { , } . The case w = 1 forces g = (6 10 4)(9 8 1)(2 7 3) · · · , g = (10 5)(4 9)(1 7)(6 3) · · · . Conclude: h g , γ ∗∞ i stabilizes O ∪ O . If w = 11, g = (6 10 z )(9 8 11) · · · , g =(10 5)( z · · · . In turn, this forces z = 10 and 10 appears twice in g .5.2.4. Proof of (5.3c) . From (5.3a) and (5.3b), conjugate by U for the first partof (5.3c). We must show x = 1 , x = 5 is false. If this does hold, then g = (6 5 y )(2 4 z )(3 w ?) · · · , g = (1 4)( y w )(3 z ) · · · . Note that y = 2, and 2 and 3 appear in distinct cycles in g using h γ ∗∞ , g i istransitive. Here is the approach for the rest: Try each case where y, z, w are in O .Whichever of { y, z, w } we try, with no loss take this to be 11.Suppose y = 11. Then, g γ ∗∞ fixes 11, contrary to g not fixing it. Now suppose z = 11. This forces w = 13 and γ ∗∞ g fixes 12, though g does not. Finally, suppose w = 11, and get an analogous contradiction to that for y = 11. With y, z, w ∈ O we have (3 w
11) a 3-cycle of g . This forces z = 13, a contradiction. Listing the cases with x = 5 , x = 1 . With these hypotheses: g = (2 1 y )(6 10 w ) · · · , g = (10 5)( y w · · · . We check that y, w O : y = 3 (or w = 3) to get simple contradictions. Example: w = 3 forces (6 2) in g ; forcing y = 7, and h g , γ ∗∞ i is not transitive. Now checkthat y O , but w ∈ O . Our normalization for being in O allows y = 11: g = (2 1 11)(6 10 w )(12 4 u )(3 13 v ) , g = (10 5)(11 4)( w · · · . So, w is 7 or 8. The first forces γ ∗∞ g to fix 6, the second forces g γ ∞ to fix 9.Now consider y, w ∈ O . If y = 7 and w = 8, then g γ ∗∞ fixes 9. If y = 8 and w = 7, then γ ∗∞ g fixes 6. We’re almost done: Try y ∈ O , and w = 11. Then y = 7 or 8. Try y = 7: g = (2 1 7)(6 10 11)(3 z ?) · · · , g = (10 5)(7 4)(11 9)( z · · · .This forces z = 6, putting z in g twice. So, y = 8: You find that from this start, g = (2 1 8)(6 10 11)(12 9 4)(13 3 7) , g = (10 5)(8 4)(11 9)(3 12)(2 7)(6 13)is forced, concluding that there is one element in Ni( A , C ∗ ) ′ .It is easy that A has no PSL ( Z ) quotient. The b-fine moduli is from Q ′′ acting faithfully on the location of the 13-cycle conjugacy class (as with n = 7).That X ∞ does not have fine moduli follows from γ ∗ and γ ∗ having fixed points.
6. Projective systems of Nielsen classes
Let F = h x , x i be the free group on two generators. Consider two simplecases for a group H acting faithfully on F , H = Z / n = 2) and H = Z / n = 3).(6.1a) The generator of H acts as x i x − i , i = 1 , ∈ Z / H acts as x x − and x x x − .We explain why these cases contrast extremely in achievable Nielsen classes.Let C be four repetitions of the nontrivial conjugacy class of H . Similarly, C ± is two repetitions of each nontrivial H class. Refer to C as p ′ conjugacy classes ifa prime p divides the orders of no elements in C . Example: C ± are 2 ′ classes. Assume G ∗ → G is a groupcover, with kernel a p group. Then p ′ conjugacy classes lift uniquely to G ∗ [ Fri95b ,Part III]. This allows viewing C as conjugacy classes in appropriate covering groups.Let P be any set of primes. Denote the collection of finite p group quotients of F , with p P , by Q F ( P ). Denote those stable under H by Q F ( P, H ). Considerinner Nielsen classes with some fixed C , P containing all primes dividing orders ofelements in C , and groups running over a collection from Q F ( P, H ): N H = { Ni( G, C ) in } { G = U × s H | U ∈ Q F ( P,H ) } . Only P = P n = { n } for n = 2 , p , G p,I = { U i } i ∈ I is a projective subsequence of (distinct) p groups from Q F ( P, H ). Form a limit group G p,I = lim ∞← i U i × s H . Assumefurther, all Nielsen classes Ni( U i × s H, C ) are nonempty. Then, { Ni( U i × s H, C ) in } i ∈ I forms a project system with a nonempty limit Ni( G p,I , C ). If C has r entries, thenthe Hurwitz monodromy group H r = h q , . . . , q r − i naturally acts (by (3.3)) on anyof these inner (or absolute) Nielsen classes. We use just r = 4. Problem . Assume I is infinite. What are the maximal groups G p,I fromwhich we get nonempty limit Nielsen classes Ni( G p,I , C )? ENUS 0 PROBLEMS 15
We call such maximal groups C p -Nielsen class limits. Prob. 6.7 refines this.For r = 4, consider maximal limits of projective systems of reduced componentsthat have genus 0 or 1. Allow | I | here to be bounded. The strong Conjecture onModular Towers [ FrS04 , § n = 3 to say this. Each such sequenceshould be bounded and there should be only finitely many (running over all p P ).Such genus 0 or 1 components have application. Ex. 6.2 is one such.Any profinite pro- p group ˆ P ′ has a universal subgroup generated by p th powersand commutators from ˆ P ′ . This is the Frattini subgroup, denoted Φ( P ′ ). The k thiterate of this group is Φ k ( P ′ ). § { ¯ H rd k,p } k ≥ . Theseare the spaces for the Nielsen classes Ni( ˆ F ,p / Φ k ( ˆ F ,p ) × s H , C ± ) in , rd . Example . Let ϕ : G ( A ) → A be the universal exponent 2 extensionof A . We explain: If ϕ : G ∗ → A is a cover with abelian exponent 2 group askernel, then there is a map ψ : G ( A ) → G ∗ with ϕ ◦ ψ = ϕ . The space H rd2 , hassix components [ BFr02 , Ex. 9.1, Ex. 9.3]. Two have genus 0, and two have genus1. Let K be a real number field. Then, there is only one possibility for infinitelymany (reduced equivalence classes of) K regular realizations of G ( A ) with fourbranch points. It is that the genus 1 components of ¯ H rd2 , have definition field K and infinitely many K rational points. The genus 0 components here have no realpoints. [ FrS04 ] explains this in more detail. N = { Ni( G, C ) in } G ∈ Q F (2) × s H . The first sentence of Prop. 6.3 re-states an argument from [
Fri95b , § Achievable Nielsen classes from modular curves.
Let zzz = { z , . . . , z } beany four distinct points of P z , without concern to order. As in § P z \ zzz = U zzz .This group identifies with the free group on four generators σσσ = ( σ , . . . , σ ),modulo the product-one relation σ σ σ σ = 1. Denote its completion with respectto all normal subgroups by ˆ F σσσ . Let ˆ Z p (resp. ˆ F ,p ) be the similar completion of Z (resp. F ) by all normal subgroups with p group quotient. Proposition . Let ˆ D σσσ be the quotient of ˆ F σσσ by the relations σ i = 1 , i = 1 , , , so σ σ = σ σ ) . Then, Q p =2 ˆ Z p × s H ≡ ˆ D σσσ . Also, ˆ Z p × s H is the unique C p -Nielsen class limit. Proof.
We show combinatorially that ˆ D σσσ is ˆ Z × s H and that σ σ and σ σ are independent generators of ˆ Z . Then, σ is a generator of H which we regard as {± } acting on ˆ Z by multiplication. First: σ ( σ σ ) σ = σ σ shows σ conjugates σ σ to its inverse. Also,( σ σ )( σ σ ) = ( σ σ ) σ ( σ σ ) σ = ( σ σ )( σ σ )shows the said generators commute. The maximal pro- p quotient is Z p × s {± } .We have only to show Nielsen classes with G = U × s H , and U an abelianquotient of Z , are nonempty. It suffices to deal with the cofinal family of U s,( Z /p k +1 ) , p = 2. § (cid:3) Remark . Any line (pro-cyclic group) in Z p produces a dihedral group D p ∞ from multiplication by -1 on this line. Nonempty Nielsen classes in Prop. 6.3. In G p k +1 = ( Z /p k +1 ) × s {± } , { ( − vvv ) | vvv ∈ ( Z /p k +1 ) } are the involutions. Write vvv = ( a, b ), a, b ∈ Z /p k +1 . Themultiplication ( − vvv )( − vvv ) yields (1; vvv − vvv ) as in the matrix product (cid:0) − vvv (cid:1)(cid:0) − vvv (cid:1) . We have an explicit description of the Nielsen classes Ni( G p k +1 , C ). Elementsare 4-tuples (( − vvv ) , . . . , ( − vvv )) satisfying two conditions from § vvv − vvv + vvv − vvv ; and(6.2b) Generation: h vvv i − vvv j , ≤ i, j ≤ i = ( Z /p k +1 ) .Apply conjugation in G p k +1 to assume vvv = 0. Now take vvv = (1 , vvv = (0 , vvv from the product-one. This shows the Nielsen class is nonempty.To simplify our discussion we have taken inner Nielsen classes. What really makesan interesting story is the relation between inner and absolute Nielsen classes. Usethe natural inclusion G p k +1 ⊳ ( Z /p k +1 ) × s GL ( Z /p k +1 ) regarding both groups aspermutations of ( Z /p k +1 ) .A general theorem in [ FV91 ] applies here. It says the natural map from H ( G p k +1 , C ) in → H ( G p k +1 , C ) abs is Galois with group GL ( Z /p k +1 ) / {± } .We give our second proof for nonempty Nielsen classes to clarify the applica-tion of [ Se68 , IV-20]. This shows, depending on the j -value of the 4 branchpoints for the cover ϕ ppp : X ppp → P z , we can say explicit things about the fiberof H ( G p k +1 , C ) in → H ( G p k +1 , C ) abs over ppp ∈ H ( G p k +1 , C ) abs .We will now display the cover ϕ ppp . Let E be any elliptic curve in Weierstrassnormal form, and [ p k +1 ] : E → E multiplication by p k +1 . Mod out by the actionof {± } on both sides of this isogeny to get E/ {± } = P wϕ pk +1 −→ E/ {± } = P z , a degree p k +1) rational function. Compose E → E/ {± } and ϕ p k +1 for the Galoisclosure of ϕ p k +1 . This geometrically shows Ni( G p k +1 , C ) = ∅ . If E has definitionfield K , so does ϕ p k +1 . We may, however, expect the Galois closure field of ϕ p k +1 to have interesting constants, from the definition fields of p k +1 division points on E . This is the subject of Serre’s Theorem and [ Fr04 , § N = { Ni( G, C ± ) in } G ∈ Q F (3) × s H . Prop. 6.5 says, unlike n = 2, forany G ∈ N , Ni( G, C ± ) in is nonempty. This vastly differs from the conclusion ofProp. 6.3. Our proof combines Harbater-Mumford reps. (Ex. 6.6) with the cyclic action of H on ( Z /p ) . We make essential use of the Frattini property. Proposition . ˆ F ,p × s H is the unique C ± p -Nielsen class limit. Proof.
First we show Nielsen classes with G = G p = ( Z /p ) × s H ( p = 2) arenonempty by showing each contains an H-M rep. Let h α i = Z /
3, with the notationon the left meaning multiplicative notation (so, 1 is the identity). The action of α is from (6.1b). For multiplication in G p use the analog of that in § g = ( α, vvv ) , g = ( α, vvv ) ∈ G generate G . Then, ( g , g − , g , g − ) isin Ni( G, C ± ) in . Further this element is an H-M rep. Conjugate this 4-tuple by(1 , vvv ) (for inner equivalence) to change g to ( α, vvv + vvv − α ( vvv )). As I − α isinvertible on ( Z /p ) we can choose vvv so vvv = 000. To find such generators, consider g g − = (1 , − vvv ) and g g = (1 , α − ( v )). So, g , g are generators precisely if ENUS 0 PROBLEMS 17 h− vvv , α − ( v ) i = ( Z /p ) . Such a vvv exists because the eigenvalues of α are distinct.So ( Z /p ) is a cyclic h α i module. If x + x + 1 mod p has no solution, then h α i actsirreducibly and any vvv = 000 works. From quadratic reciprocity this is equivalent to − p , or p ≡ − N defined by G = U × s H with U having( Z /p ) as a quotient. There is a surjective map ψ : G → ( Z /p ) × s H , and it is aFrattini cover. So, if g ′ , g ′ are generators of ( Z /p ) × s H given by the proof above,then any respective order 3 lifts of g ′ , g ′ to g , g ∈ G will automatically generate G . Therefore the representative ( g , g − , g , g − ) of the Nielsen class Ni( G, C ± ) in lifts ( g ′ , ( g ′ ) − , g ′ , ( g ′ ) − ). This shows all Nielsen classes are nonempty. (cid:3) Let r = 4 and ggg ∈ Ni( G, C ) in , rd (or any reducedequivalence). Then, the orbit of ggg under the cusp group Cu = h q , Q ′′ i interpretsas a cusp of the corresponding j -line cover. Certain Nielsen class representativesdefine especially useful cusps. We define these relative to a prime p . [ FrS04 , § p ′ cusps, and when r = 4, their representatives ggg = ( g , . . . , g ) inreduced Nielsen classes have the following property:(6.3) Both H , ( ggg ) def = h g , g i and H , ( ggg ) = h g , g i are p ′ groups; theirorders are prime to p .Recall the shift sh = q · · · q r − as an operator on Nielsen classes: For ggg ∈ Ni( G, C ) , ( ggg ) sh = ( g , . . . , g r , g ) . Example . Suppose h and h are elements generating a group G and consider the 4-tuple hhh = ( h , h − , h , h − ). If hhh ∈ Ni( G, C ), we say hhh is a Harbater-Mumford representative (H-M rep.) of the Nielsen class. If C are p ′ conjugacy classes, then ( hhh ) sh is a a representative for a p ′ cusp.[ FrS04 , Prop. 5.1] generalizes how we use H-M reps., in Prop. 6.5 as follows.Suppose level 0 of a Modular Tower, for the prime p , has representatives of g- p ′ cusps. Then applying Schur-Zassenhaus gives projective systems of g- p ′ cusps. Thisis the only method we know to show there are nonempty Nielsen classes at all levels.So, we don’t know whether g- p ′ cusps are necessary for a result like Prop. 6.5.For this case alone we pose a problem that aims at deciding that. Use the actionof H = h q , q , q i in § N , it is easy toshow all representatives of g- p ′ cusps ( p = 3) must be shifts of H-M reps. Problem . Suppose O is an H orbit on Ni( ˆ F ,p × s H , C ± ) in , rd . Must O contain the shift of an H-M rep.?
7. Relating Problem g =0 and Problem g =0 Our last topic discusses, using the work of others, what might be the meaningand significance of an intrinsic uniformizer for a 1-dimensional genus 0 moduli space. As § Fr95a , p. 349]and [
Fr99 , §
8] in 1969 at the writing of [
Fr73 ]. Precisely relating to the j -lineis more recent, though you see these computations are easy. Further, they createa framework for finding a parameter for a moduli cover of the j -line having aninternal interpretation. Couveignes calculations.
Jean-Marc Couveignes used computer-assistedcalculations to find equations for Davenport pairs [
CoCa99 ]. He also gave a tech-nique for uniformizing some moduli of the type we are considering. Necessarily, themoduli was of genus 0 covers, and the moduli space of genus 0 [
Co00 ]. Prop. 4.1and Prop. 5.1 produce a natural geometry behind the genus 0 moduli from Daven-port’s arithmetically defined problem. We compare our computation with that from[
CoCa99 ], whose tool was
PARI Ver. 1.920.24 with
Maple used as a check.Equations for those degree 7, 13 and 15 polynomials are in [
CoCa99 , § CoCa99 , § CoCa99 , § Fr80 , p. 593] has Birch’s brute force calcu-lation of degree 7 Davenport pairs. He and Guy also did the degree 11 case (threebranch point covers, so up to reduced equivalence this is just one pair of polynomi-als). Degree 13 is the most interesting. There the difference set argument producesthe nontrivial intertwining of four polynomials for each point of the space X ∞ . Asthe [ CoCa99 ] calculations take considerable space, we don’t repeat them. Still,a statistical comparison indicates the extra complexity (supported by theory) inthe degree 13 case. Here is the character count for Couveignes’ expression for thegeneral polynomial in each case (not counting spaces):(7.1) degree 7: 146 ; degree 13: 1346 ; degree 15: 819 .7.1.2.
Significance of the j -line cover. Neither Birch nor Couveignes relatestheir equations to the j -line. Still, two points help compare [ Co00 ] with our goals.(7.2a) Using [
GHP88 ], Harbater patching can sometimes produce the equa-tion for the general member a family of genus 0 covers.(7.2b) Using formal fibers of a moduli problem avoids direct computation ofa possibly large base extension (when no version of rigidity holds).I think (7.2a) refers to moduli of genus 0 covers, when a Nielsen class gives datato normalize a parameter for each cover in the family. We explain below thiscomputational handle that starts from degenerate situations and at the cusps.Grothendiecks’ original method, reviewed in [
Fr99 , § Artinapproximation to achieve an algebraic deformation. It is around that last step thatCouveignes uses his genus 0 assumptions.The example of [
CoCa99 , p. 48] has some version of reduced parameters,though over the λ -line, not the j -line. [ CoCa99 , p. 56] gives the practical sense of(7.2b). We use his formal deformation parameter µ . His example serves no exteriorproblem. Still, it illustrates the computational technique. Though we admire it,we want to show why there are many constraints on its use to compute equations.7.1.3. Constraints in Couveignes’ example.
The group for his Nielsen classes is S , and his conjugacy classes C c are those represented by the entries of(7.3) ggg µ = ((2 3 4)(6 7) , (1 2 5 6) , (1 7) , (2 3 4 5 6 7) − ) . He interpolates between two 3 branch point covers with respective branch cycles:(7.4) ggg = ((2 3 4)(6 7) , (1 2 5 6) , (1 2 3 4 5 6 7) − ) and; ggg = ((1 2 3 4 5 6 7) , (1 7) , (2 3 4 5 6 7) − ) . His four conjugacy classes are all distinct. Then, the reduced family of absolutecovers in the Nielsen class, as a j -line cover, automatically factors through the λ -line as ¯ ψ c : ¯ H ( S , C c ) abs , rd → P λ . He finds explicit equations ϕ and ϕ for the unique covers branched over { , , ∞} ∈ P j with branch cycles represented by ggg and ggg . An analyst would view this as forming ϕ µ : P w µ → P z as a function of µ ∈ (0 ,
1) so that as µ ϕ µ degenerates to ϕ (resp. ϕ ).Topologically this is a coalescing , respectively, of the first two (last two) branchpoints. Realize, however, compatible with the statement in § ϕ µ in the coordinates of the parameterspace ¯ H ( S , C c ) abs , rd . [ CoCa99 , p. 43-48] refers to [
DFr94 , Thm. 4.5] to computethe action of the λ -line (not j -line) version of γ ∗ , γ ∗ , γ ∗∞ (as in Propositions 4.1and 5.1). He concludes ¯ H ( S , C c ) abs , rd has genus 0. Dropping reduced equivalencegives a fine moduli space (reason as in § λ -line, we must express coordinates for ppp ∈ H ( S , C c ) abs , rd in thealgebraic closure of Q ( λ ). This space is irreducible, from the analog of γ , γ , γ ∞ acting transitively. For the same reason as in our Davenport pair examples, reducedequivalence gives a b-fine moduli space ( § braid rigidity . So, the cardinality of the reduced Nielsen class equalsthe degree of those coordinates over Q ( λ ) ([ Fr77 , Cor. 5.3], [
MM99 , Thm. 5.3] or[
V¨o96 , § Co00 , p. 50] uses a set of classical generators (as in § µ w µ (to appear in ϕ µ ( w µ )) by selecting 3distinguished points on the cover ϕ µ . These correspond to selecting 3 distinguisheddisjoint cycles in the branch cycles ggg µ in (7.3). He wants w µ = w µ, to survive (inthe limit µ
1) to give w (similarly with w ). So, the points he chooses mustsurvive the coalescing. It is a constraint on the explicit equations that he can pickthree such points. It is a separate constraint that he can do the same on the otherlimit µ
0, producing a parameter w µ, for µ near 0.Here in [ Co00 , p. 50-51] a reader might have difficulty (see the reference).Naming the three points (labeled V , V , V ), where w µ, takes respective values0 , , ∞ (chosen for µ
1) appears after their first use in equations. That is becauseof a misordering of the printed pages. He analytically continues these choices along µ ∈ (0 , w µ, /w µ, is w µ, ( V ) /w µ, ( V ), expressed in meaningfulconstants from ramification of ¯ ψ c , as a function of µ . [ Co00 ] then explains whatto do with expressions for ϕ µ ( w µ,k ), k = 0 , precisely express coefficients of a general member of the family.7.1.4. Using geometric compactifications.
Couveignes applied coordinates from[
GHP88 ] for his explicit compactification. Many use a compactification aroundcusps, placing over the cusp something called an admissible cover of (singular)curves. Arithmetic applications require knowing that the constant field’s absoluteGalois group detects the situation’s geometry. My version is a specialization se-quence [ Fri95b , Thm. 3.21]. This gives meaningful action on projective sequencesof cusps. The goal is to see exactly how G Q acts from its preserving geometriccollections like g- p ′ cusps ( § DDE04 , Thm. 1.3and Thm. 4.1] gave a treatment of [
Fri95b , Thm. 3.21] using admissible covers (theversion in [
We99 ]). For that reason, they compactify with a family that has
H-M admissible covers around the H-M cusps. (They use Hurwitz, not reduced Hurwitz,spaces.) A corollary of [
Fri95b , Thm. 3.21] has simple testable hypotheses thatguarantee there is a unique component of the moduli space containing H-M cusps(so it is over Q ). Those hypotheses rarely hold if r = 4.So, I wish someone could do the following. Problem . Approach the genus 1 components in Ex. 6.2 as did Couveignesfor his examples.Couveignes’ explicit constrains fail miserably here. Ex. 6.2 is a family of veryhigh genus curves with no distinguished disjoint cycles in their branch cycle de-scriptions. Still, the topics of [
DDE04 ] and [
We99 ] are relevant. The two genus 1components of Ex. 6.2 are both
H-M rep. components. The two components comefrom corresponding orbits for H acting on inner Nielsen class orbits. They are,however, not total mysteries: An outer automorphism of G ( A ) joins the orbits.(None of that is obvious; [ FrS04 , §
7] will have complete documentation.)7.1.5.
An intrinsic uniformizing parameter.
Interest in variables separated poly-nomials f ( x ) − g ( y ), those § DLS61 ] and [
DS64 ]). As [
Fr04 , § X n ( n = 7 , ,
15) using the arithmetic behind their investigation.We don’t know if there is such, though [
Fr04 ] suggests some based on functions in( x, y ) (satisfying f ( x ) − g ( y ) = 0) that are constant along fibers to the space X n . I speak only of John’s in-fluence in specific mathematical situations related to this paper. John and I had aconversation in 1986 on the way to lunch at University of Florida. This one conver-sation brings a luminous memory, so singular it may compress many conversations.7.2.1.
John’s formation of Problem g =0 . As we walked, I summarized the grouptheory from solving several problems, like Davenport’s. Each came with an an equa-tion that we could rewrite as a phrase on primitive (genus 0) covers. My conclusion:There were always but finitely many rational function degree counterexamples tothe most optimistic hopes. Yet, there were some counterexamples.Further, to solve these problems required nonobvious aspects of groups. Forexample: In Davenport’s problem, we needed difference sets and knowledge of allthe finite groups with two distinct doubly transitive permutation representationsthat were equivalent as group representations (I got this from [
CKS76 ]). Quitelike the classification, many simple groups and some new number theory, impingedon locating the polynomial Davenport pairs. That was my pitch.John responded that he was seized with the underlying problem. His initialformulation was this. Suppose G is a composition factor of the covering monodromyof ϕ : X → P z with X of genus g . Then, it is a genus g group. Problem . We fix an integer g and exclude alternating and cyclic groups.Show: Only finitely many simple groups have genus g .As above, the genus 0 primitive cover case suffices. This was my encouragementfor the project. It is nontrivial to grab a significant monodromy group at random.(You will always get the excluded groups.) Still, the genus 0 problem would displayexceptions. These should contribute conspicuously, as happened with the Schur, ENUS 0 PROBLEMS 21
Davenport and HIT problems. That is, a general theorem would have sporadiccounterexamples. While they might be baffling, they would nevertheless add tothe perceived depth of the result. Especially they would guide situations of highergenus, and in positive characteristic. Example: I suggested there would be newprimitive rational functions beyond those coming from elliptic curves, that had the
Schur cover property : Giving one-one maps on P z ( F p ) for ∞ -ly many primes p .John suggested we work toward this immediately. I had hoped, then, he wouldbe interested in my approach to using the universal p -Frattini cover of a finitegroup. My response was that Bob Guralnick enjoyed this type of problem andknew immensely more about the classification than I. So was born Problem g =0 , andthe collaboration of Guralnick-Thompson.7.2.2. Progress on Problem g =0 . John showed me the initial list from his workwith Bob on the affine group case. The display mode was groups presented bybranch cycle generators: an absolute Nielsen class ( § branch cycle argument [ Fr77 , p. 62] to seethey had the Schur cover property. (You measure this by distinguishing between itsarithmetic and geometric monodromy groups.) meant it did not come from ellipticcurves, or twists of cyclic or Chebychev polynomials.We now have a list of Nielsen classes sporadic for the Schur property (Schur-sporadic; [
GMS03 , Thm. 1.4]). [
Fr04 , § Problem g =0 in John’s form is true . There are only finitely many sporadic genus0 groups. Most major contributors are in this chronological list: [ GT90 ], [
As90 ],[
LS91 ], [
S91 ], [
GN92 ], [
GN95 ], [
LSh99 ], and [
FMa01 ]. Reverting to the primi-tive case parceled the task through the 5-branch Aschbacher-O’Nan-Scott classifi-cation of primitive groups [
AS85 ].7.2.3.
Guralnick’s optimistic conjecture.
Yet, there is an obvious gap betweenthe early papers and the two at the end. The title of [
LSh99 ] reveals it did notlist examples precisely as John did at the beginning. We can’t yet expect the exceptional
Chevalley groups to fall easily to such explicitness; you can’t grab yourfavorite permutation representation with them. Still, composition factors are onething, actual genus 0 primitive monodromy groups another.
Definition . We say T : G → S n , a faithful permutation representation,with properties (7.5) and (7.6) is 0 -sporadic .Denote S n on unordered k sets of { , . . . , n } by T n,k : S n → S ( nk ) by ( T n, thestandard action). Alluding to S n (or A n ) with T n,k nearby refers to this presenta-tion. In (7.6), V a = ( Z /p ) a ( p a prime). Use § T V a case on points of V a ; C can be S . For the second ( A n , T n, ) case, T : G → S n .(7.5) ( G, T ) is the monodromy group of a primitive ( § ϕ : X → P z with X of genus 0.(7.6) ( G, T ) is not in this list of group-permutation types. • ( A n , T n, ): A n ≤ G ≤ S n , or A n × A n × s Z / ≤ G ≤ S n × S n × s Z / • ( A n , T n, ): A n ≤ G ≤ S n . • T V a : G = V × s C , a ∈ { , } , | C | = d ∈ { , , , , } and a = 2 only if d does not divide p − f ∈ C ( x ) represent 0-sporadic groups by f : P x → P z . Wesay ( G, T ) is polynomial 0-sporadic , if some f ∈ C [ x ] represents it. Definition . Similarly, we say (
G, T ) is g -sporadic if (7.5) holds replacinggenus 0 by genus g .For g -sporadic, the list of (7.6) is too large. [ GMS03 , Thm. 4.1] tips off theadjustments for g = 1 ( § g > g -sporadic groups should be just A n ≤ G ≤ S n of T n, type, and cyclic or dihedral groups.[ GSh04 ] has 0-sporadics with an A n component. [ FGMa02 ] has 0-sporadicsgroups with a rank 1 Chevalley group component. Magaard has written an outlineof the large final step: Where components are higher rank Chevalley groups. Likethe classification itself, someone going after a concise list of such examples for aparticular problem will have difficulty culling the list for their problem. Variouslists of the 0-sporadics appear in many papers.If someone outside group theory comes upon a problem suitable for the mon-odromy method or some other, can they go to these papers, look at the lists andfinish their projects? [
So01 ] has anecdotes and lists on the classification that manynon-group theorists can read. Pointedly, however, is it sufficient to allow you orI to have replaced any contributor to [
GMS03 ]? Unlikely! How about to read[
GMS03 ]? Maybe! Yet, not without considerable motivation.One needs familiarity with the relation between primitive subgroups of S n andsimple groups, the description from Aschbacher-O’Nan-Scott. That does not relyon the classification. Rather, it treats simple group appearances as a black box.To decide if there are simple groups satisfying extra conditions contributing to theappearance of a particular primitive group, you must know special informationabout the groups in [ So01 , p. 341].7.2.4.
Qualitative versus quantitative.
John’s desire for documenting 0-sporadicgroups added many pages to the literature. What did particular examples do? Howdoes one present specific examples to be useful? We have been suggesting § FGS93 ] and [
GMS03 ], usea condition about a group normalizing G . This eliminated much of the primitivegroup classification. A seeker after applying the same method may find they, too,have such a useful condition, making it unnecessary to rustle through many of thelists like [ FGMa02 ] or [
GSh04 ]. We explain. Warning: An ( S , T , ) sporadiccase occurs in answer a simple question about all indecomposable f ∈ Z [ x ] onHilbert’s irreducibility theorem. Nor can you just look at a 0-sporadic Nielsen classto decide if it has an arithmetic property to be HIT-sporadic (a name coined forthis occasion; [ DFr99 , § § GMS03 , Thm. 1.4] classified 0-sporadics with the Schur property (Schur-sporadics) over number fields. In [
Fr80 , p. 586] we used the Schur property toshow how to handle an entwining between arithmetic and geometric monodromygroups. Group theory setup: Two subgroups G ≤ ˆ G ≤ S n of S n have this property.(7.7) There is a τ ∈ ˆ G \ G so that g in the coset Gτ implies g fixes preciselyone integer (see Ex.7.5).[ GMS03 ] calls our Schur covering property, arithmetic exceptionality.)[
GMS03 , Thm. 1.4, c)] has the list where the genus of the Galois closureexceeds 1. These are the Schur-sporadics: Only finitely many 0-sporadic groupsoccur. Yet, John’s original problem posed sporadic to mean the composition factorsincluded other than cyclic or alternating groups. All three types of degree 25 alluded
ENUS 0 PROBLEMS 23 to above were not sporadic from this criterion. Indeed, the only nonsporadic fromthis criterion in the whole list were the groups PSL ( n ) with n = 8 and 9, andthese had polynomial forerunners from [ Mu95 ]. The more optimistic conjecture of § big theorem when you hear of a study of dihedral groups? Yet, [ Se68 ] is a big theorem. Itis the arithmetic of special four branch point dihedral covers. The kind we call involution : They are in the Nielsen class Ni( D p k +1 , C ), four repetitions of theinvolution conjugacy class in D p k +1 . [ Fr04 , § Fr04 , App. C] shows alternating anddihedral groups are dual for arithmetic questions about monodromy group covers.
Example . Notice that D p ≤ Z /p × s ( Z /p ) ∗ ≤ S p satisfies the criterion of (7.7). The goal is to describerational functions f ∈ K ( x ) with K some number field so the Galois closure groupˆ G f (resp. G f ) of f ( x ) − z over K ( z ) (resp. C ( z )) gives such a pair. The only sim-plification is that f can’t decompose into lower degree polynomials over K . When E is an elliptic curve without complex multiplication in § f indecom-posable over K , but decomposable over C . This is one of the two nonsporadic caseswhere the Galois closure cover for f : P w → P z has genus 1. [ GMS03 , Thm. 1.4,b)] has the complete list where the Galois closure cover has genus 1. A reader newto this will see some that look sporadic. Yet, those come from elliptic curves andtopics like complex multiplication. They are cases where K = Q and there arespecial isogenies over Q defined by a p -division point not over Q .7.2.5. Monstrous Moonshine uniformizers.
Recall a rough statement from
Mon-strous Moonshine . Most genus 0 quotients from modular subgroups of PSL ( Z )have uniformizers from θ -functions that are automorphic functions on the upper halfplane. This inspired conjecture, to which John significantly contributed [ To79a ],gives away John’s intense desire to see the genus 0 monodromy covers group theo-retically. A recent Fields Medal to Borcherds on this topic corroborates the world’sinterest in genus 0 function fields, if the uniformizer has significance .The Santa Cruz conference of 1979 alluded to on [
So01 , p. 341] ([
Fe80 ] and[
Fr80 ] came from there) had intense discussions of Monstrous Moonshine. Thiswas soon after a suggestion by A. Ogg: He noticed that primes p dividing the orderof the Monster (simple group; denote it by M ) are those where the function fieldof the normalizer of Γ ( p ) in PSL ( R ) has genus zero. A. Pizer was present: Hecontributed that those primes satisfy a certain conjecture of Hecke relating modularforms of weight 2 to quaternion algebra θ -series [ P78 ]. Apparently Klein, Fricke andHecke had recognized the problem of finding the function field generator of genus0 quotients of the upper half-plane, not necessarily given by congruence subgroupsof PSL ( Z ). It seems somewhere in the literature is the phrase genus 0 problem attached to a specific Hecke formulation.Thompson concocted a relation between q = e πiτ -expansion coefficients of j ( τ ) = q − + P ∞ n =0 u n q n ( u = 744, u = 196884, u = 21493760, u = 864299970, u = 20245856256, u = 333202640600) and irreducible characters of M . At thetime, the Monster hadn’t been proved to exist, and even if it did, some of itscharacter degrees weren’t shown for certain. ([ To79a ] showed if the Monster ex-isted, these properties uniquely defined it.) John noted the coefficients listed for j are sums of positive integral multiples of these. With this data he conjectured a q -expansion with coefficients in Monster characters with these properties [ To79b ]: • the q expansion of j is its evaluation at 1; and • the other genus 0 modular related covers have uniformizers from itsevaluation at the other conjugacy classes of M .[ Ra00 , p. 28] discusses those automorphic forms on the upper half plane withproduct expansions following Borcherd’s characterization. Kac-Moody algebrasgive automorphic forms with a product expansion. The construction of a MonsterLie Algebra that gave q -expansions matching that predicted by Thompson is what— to a novice like myself — looks like the main story of [ Ra00 ].Is there any function, for example on any of the genus 0 spaces from this paperthat vaguely has a chance to be like such functions? Between [
BFr02 , App. B.2]and [
Ser90b ] one may conclude the following discussion.All components of the H rd p,k s in § θ -nulls canonically attached to theirmoduli definition. So do many of the quotients between H rd p,k +1 and H rd p,k . For the H rd p,k s we really do mean θ -nulls defined by analytically continuing a θ function onthe Galois cover ϕ ppp : X ppp → P z attached to ppp ∈ H rd p,k , then evaluating it at theorigin. Usually such θ -nulls have a character attached to them. Here that wouldbe related to the genus of X ppp . So, we cannot automatically assert these θ -nulls areautomorphic on the upper half plane (compare with genus 1 versions in [ FaK01 ]).[
Si63 ] (though hard to find) presents the story of θ -functions defined by unimodularquadratic forms. These do define automorphic functions on the upper half plane.The discussion for H , alluded to two genus 1 and two genus 0 components.The θ for the genus 0 components is an odd function. So its θ -null will be identically0. For the genus 1, components, however, it is even, and both those componentscover (by a degree 2 map) a genus 0 curve between H , and H , . That is onespace we suggest for significance.7.2.6. Strategies for success. [ B03 ] has a portrait of Darwin as a man of con-siderable self-confidence, one who used many strategies to further his evolutiontheory. Though this is contrary to other biographies of Darwin, the case is convinc-ing. Darwin was a voluminous correspondent, and his 14,000 + letters are recordedin many places. Those letters reflected his high place in the scientific community.They often farmed out to his correspondent the task of completing a biologicalsearch, or even a productive experiment. So, great was Darwin’s reputation thathis correspondents allowed him to travel little (in later life), and yet accumulategreat evidence for his mature volumes. Younger colleagues (the famous ThomasHuxley, for example) and even his own family (his son Francis, for instance) pre-sented him a protective team and work companions. The success of the theory ofevolution owes much to the great endeavor we call Charles Darwin.[ Ro03 ] reflects on the different way mathematical programs achieve success inhis review of essays that touch on the growth of US mathematics into the inter-national framework. He suggests the international framework that [
PaR02 ] toutsmay not be the most compelling approach to analyzing mathematical success.Recently, historians of science have tried to understand how. . . locally gained knowledge produced by research schoolsbecomes universal , a process that involves analyzing all thevarious mechanisms that produce consensus and support
ENUS 0 PROBLEMS 25 within broader scientific networks and communities. Sim-ilar studies of mathematical schools, however, have beenlacking, a circumstance . . . partly due to the prevalent be-lief that mathematical knowledge is from its very inceptionuniversal and . . . stands in no urgent need to win converts.There are two genus 0 problems: Problem g =0 and Problem g =0 . They seem verydifferent. Yet, they are two of the resonant contributions of John Thompson, outsidehis first area of renown. His influence on their solutions and applications is so large,you see I’ve struggled to complete their context. The historian remarks intrigue mefor it would be valuable to learn, along their lines, more about our community. References [AFH03] W. Aitken, M. Fried and L. Holt,
Davenport Pairs over finite fields , in proof, PJM,Dec. 2003.[As90] M. Aschbacher,
On conjectures of Guralnick and Thompson , J. Algebra ,277–343.[AS85] M. Aschbacher and L. Scott,
Maximal subgroups of finite groups , J. Alg. (1985),44–80.[A57] E. Artin, Geometric Algebra , Interscience tracts in pure and applied math. , 1957.[BFr02] P. Bailey and M. Fried, Hurwitz monodromy, spin separation and higher levels of aModular Tower , in Proceedings of Symposia in Pure Mathematics (2002) editorsM. Fried and Y. Ihara, 1999 von Neumann Conference on Arithmetic FundamentalGroups and Noncommutative Algebra, August 16-27, 1999 MSRI, 79–221.[B03] J. Browne, Charles Darwin: The Power of Place , Knopf, 2003.[Ca56] R.D. Carmichael,
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The 2-transitive permutation representa-tions of the finite Chevalley groups , TAMS (1976), 1–59.[DLS61] H. Davenport, D.J. Lewis and A. Schinzel,
Equations of the form f ( x ) = g ( y ),Quart. J. Math. Oxford (1961), 304–312.[DS64] H. Davenport, and A. Schinzel, Two problems concerning polynomials , Crelle’s J. (1964), 386–391.[DDE04] P. Debes and M. Emsalem,
Harbater-Mumford components and Towers of ModuliSpaces , Presentation by M. Emsalem at Graz, July 2003, preprint Jan. 2004.[DFr94] P. Debes and M.D. Fried,
Nonrigid situations in constructive Galois theory , PacificJournal
163 (1994), 81–122.[DFr99] P. Debes and M.D. Fried,
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On a conjecture of Schur , Mich. Math. J. (1970), 41–45.[Fr73] M.D. Fried, The field of definition of function fields and a problem in the reducibilityof polynomials in two variables , Ill. J. of Math. (1973), 128–146. [Fr77] M. Fried, Fields of definition of function fields and Hurwitz families and groups asGalois groups , Communications in Algebra (1977), 17–82.[Fr78] M. Fried, Galois groups and Complex Multiplication , T.A.M.S. (1978) 141–162.[Fr80] M.D. Fried,
Exposition on an Arithmetic-Group Theoretic Connection via Riemann’sExistence Theorem , Proceedings of Symposia in Pure Math: Santa Cruz Conferenceon Finite Groups, A.M.S. Publications (1980), 571–601.[Fr95a] M. Fried, Extension of Constants, Rigidity, and the Chowla-Zassenhaus Conjecture ,Finite Fields and their applications, Carlitz volume (1995), 326–359.[Fri95b] M. D. Fried, Modular towers:
Generalizing the relation between dihedral groups andmodular curves , Proceedings AMS-NSF Summer Conference, , 1995, Cont. Mathseries, Recent Developments in the Inverse Galois Problem, 111–171.[Fr99] M.D. Fried,
Separated variables polynomials and moduli spaces
Prelude: Arithmetic fundamental groups and noncommutative algebra ,Proceedings of Symposia in Pure Mathematics, (2002) editors M. Fried and Y. Ihara,1999 von Neumann Conference on Arithmetic Fundamental Groups and Noncommu-tative Algebra, August 16-27, 1999 MSRI, vii–xxx.[Fr04] M.D. Fried, Extension of constants series and towers of exceptional cov-ers
Riemann’s existence theorem: An elementary approach to moduli
Schur covers and Carlitz’s conjecture , Israel J. (1993), 157–225.[FrS04] M.D. Fried and D. Semmen, Schur multiplier types and Shimura-like systemsof varieties
The inverse Galois problem and rational points on modulispaces , Math. Annalen (1991), 771–800.[FMa01] D. Frohardt and K. Magaard,
Composition Factors of Monodromy Groups , Annals ofMath. (2001), 1–19[FGMa02] D. Frohardt, R.M. Guralnick and K. Magaard,
Genus 0 actions of groups of Lie rank1 , in Proceedings of Symposia in Pure Mathematics (2002) editors M. Fried andY. Ihara, 1999 von Neumann Conference on Arithmetic Fundamental Groups and Non-commutative Algebra, August 16-27, 1999 MSRI, 449–483.[GHP88] L. Gerritzen, F. Herrlich, and M. van der Put, Stable n -pointed trees of projective lines ,Ind. Math. (1988), 131–163.[GN92] R.M. Guralnick, The genus of a permutation group , in Groups, Combinatorics andGeometry, Ed: M. Liebeck and J. Saxl, LMS Lecture Note Series , CUP, Longdon,1992.[GMS03] R. Guralnick, P. M¨uller and J. Saxl,
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Monodromy groups of branched coverings: thegeneric case , Proceedings AMS-NSF Summer Conference, , 1995, Cont. Math se-ries, Ed: M. Fried Recent Developments in the Inverse Galois Problem, 325–352.[GSh04] R.M. Guralnick and J. Shareshian,
Symmetric and Alternating Groups as MonodromyGroups of Riemann Surfaces I , preprint.[GT90] R.M. Guralnick and J.G. Thompson,
Finite groups of genus 0 , J. Algebra (1990),303–341.[LS91] M. Liebeck and J. Saxl,
Minimal degrees of primitive permutation groups, with anapplication to monodromy groups of covers of Riemann surfaces , PLMS (3) 63 (1991),266–314.[LSh99] M. Liebeck and A. Shalev,
Simple groups, permutation groups, and probability , 497–520.
ENUS 0 PROBLEMS 27 [MM99] G. Malle and B.H. Matzat,
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40 (2003), 535–542.[Se68] J.-P. Serre,
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Revˆetements a ramification impaire et thˆeta-caract´eristiques , C. R. Acad.Sci. Paris (1990), 547–552.[S91] T. Shih,
A note on groups of genus zero , Comm. Alg. (1991), 2813–2826.[Si63] C.L. Siegel, Analytic Zahlentheorie II Vorlesungen , gehalten im Wintersemester1963/64 an der Universit¨at G¨ottingen, mimeographed notes.[To79a] J.G. Thompson,
Finite groups and modular functions , BLMS 11 (3) (1979), 347–351.[To79b] J.G. Thompson,
Some Numerology between the Fischer-Griess Monster and the ellipticmodular function , BLMS 11 (3) (1979), 340–346.[Ra00] U. Ray,
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An extension of F. Klein’s level concept , Ill. J. Math. (1964), 529–535. Emeritus, University of California at Irvine
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